ISSN1842-6298 Volume1(2006), 61 – 69
A FUNCTIONAL CALCULUS FOR QUOTIENT
BOUNDED OPERATORS
Sorin Mirel Stoian
Abstract. If(X;P) is a sequentially locally convex space, then a quotient bounded operator T 2QP(X) is regular (in the sense of Waelbroeck) if and only if it is a bounded element (in the
sense of Allan) of algebraQP(X). The classic functional calculus for bounded operators on Banach
space is generalized for bounded elements of algebraQP(X).
1
Introduction
It is well-known that ifXis a Banach space andL(X)is Banach algebra of bounded operators onX, then formula
f(T) = 21iR
f(z)R(z; T)dz,
( where f is an analytic function on some neighborhood of (T), is a closed recti…able Jordan curve whose interior domain D is such that (T) D, and f is analytic on D and continuous on D[ ) de…nes a homomorphism f ! f(T) from the set of all analytic functions on some neighborhood of (T)intoL(X), with very useful properties.
Through this paper all locally convex spaces will be assumed Hausdor¤, over complex …eldC, and all operators will be linear. If X and Y are topological vector
spaces we denote by L(X; Y)(L(X; Y)) the algebra of linear operators (continuous operators) fromX toY.
Any family P of seminorms which generate the topology of locally convex space X (in the sense that the topology of X is the coarsest with respect to which all seminorms ofP are continuous) will be called a calibration on X. A calibration P is characterized by the property that the collection of all sets
2000 Mathematics Subject Classi…cation: 47A60, 46A03 Keywords: locally convex space, functional calculus.
S(p; ) =fx2Xjp(x)< g;(p2 P; >0);
constitute a neighborhoods sub-base at 0. A calibration on X will be principal if it is directed. The set of all calibrations for X is denoted byC(X) and the set of all principal calibration byC0(X).
If(X;P)is a locally convex algebra and each seminormsp2 P is submultiplica-tive then(X;P) is locally multiplicative convex algebra or l.m.c.-algebra.
Any family of seminorms on a linear space is partially ordered by relation „ ”, where
p q ,p(x) q(x),(8)x2X.
A family of seminorms is preordered by relation „ ”, where
p q, there exists somer >0such thatp(x) rq(x), for all x2X.
If p q and q p, we write p q.
De…nition 1. Two families P1 and P2 of seminorms on a linear space are called Q-equivalent ( denoted P1 P2) provided:
1. for each p12 P1 there existsp2 2 P2 such thatp1 p2;
2. for each p2 2P2 there exists p1 2 P1 such thatp2 p1.
It is obvious that two Q-equivalent and separating families of seminorms on a linear space generate the same locally convex topology.
De…nition 2. If(X;P), (Y;Q) are locally convex spaces, then for eachp; q2P the application mpq :L(X; Y)!R[ f1g, de…ned by
mpq(T) = sup p(x)6=0
q(T x)
p(x) ;(8)T 2L(X; Y):
is called the mixed operator seminorm ofT associated withpandq. WhenX =Y and p=q we use notation p^=mpp.
Lemma 3 ([9]). If (X;P), (Y;Q) are locally convex spaces and T 2L(X; Y), then
1. mpq(T) = sup p(x)=1
q(T x) = sup
p(x) 1
q(T x);(8)p2P;(8)q2Q;
2. q(T x) mpq(T)p(x),(8)x2X, whenever mpq(T) <1.
3. mpq(T) = inffM >0jq(T x) M p(x);(8)x2Xg, whenever mpq(T)<1.
De…nition 4. An operator T on a locally convex space X is quotient bounded with respect to a calibration P 2 C(X) if for every seminorm p 2 P there exists some cp >0 such that
The class of quotient bounded operators with respect to a calibrationP 2 C(X)
is denoted byQP(X). If X is a locally convex space andP 2C(X), then for every p2 P the application p^:QP(X)!R de…ned by
^
p(T) = inffr >0 jp(T x) r p(x);(8)x2X g;
is a submultiplicative seminorm on QP(X), satisfying p^(I) = 1. We denote by P^
the familyf P j^ p2 P g.
Lemma 5 ([8]). If X is a sequentially complete convex space, then QP(X) is a
sequentially complete m-convex algebra for allP 2 C(X).
De…nition 6. Let X be a locally convex space and T 2 QP(X). We say that
2 (QP; T) if the inverse of I T exists and ( I T) 12QP(X). Spectral sets
(QP; T) are de…ned to be complements of resolvent sets (QP; T).
Let (X;P) be a locally convex space and T 2 QP(X). We have said that T
is bounded element of the algebra QP(X) if it is bounded element in the sens of
G.R.Allan [1], i.e. some scalar multiple of it generates a bounded semigroup. The class of bounded element of QP(X) is denoted by(QP(X))0.
Proposition 7 ([5]). Let X is a locally convex space and P 2 C(X).
1. QP(X) is a unital subalgebra of the algebra of continuous linear operators on X
2. QP(X) is a unitary l.m.c.-algebra with respect to the topology determined by
^
P
3. If P02 C(X) such thatP P0, then Q
P0(X) =QP(X) and P^ P^ 0
4. The topology generated by P^ on QP(X)is …ner than the topology of uniform convergence on bounded subsets of X
De…nition 8. If (X;P) is a locally convex space and T 2 QP(X) we denote by rP(T) the radius of boundness of operator T in QP(X), i.e.
rP(T) = inff >0 j 1T generates a bounded semigroup in QP(X)g:
We have said thatrP(T) is the P-spectral radius of the operator T.
Proposition 9 ([8]). Let X be a sequentially complete locally convex space and P 2C(X). If T 2QP(X), then j (QP; T)j=rP(T):
De…nition 10. Let P be a calibration on X. A linear operator T : X ! X is universally bounded on (X;P) if exists a constant c0 >0 such that
Denote by BP(X)the collection of all universally bounded operators on (X;P).
It is obvious that QP(X) BP(X) L(X).
Lemma 11. If P a calibration on X, thenBP(X) is a unital normed algebra with respect to the norm k kP de…ned by
kTkP = inffM >0 j p(T x) M p(x);(8)x2X;(8) p2 Pg:
Corollary 12. If P 2 C(X), then for each T 2BP(X) we have
kTkP = supfmpp(T)jp2 Pg;(8)T 2BP(X):
Proposition 13 ([5]). Let X be a locally convex space and P 2 C(X). Then:
1. BP(X) is a subalgebra of L(X);
2. (BP(X);k kP) is unitary normed algebra;
3. for each P0 2 C(X) with the propertyP P0, we have
BP(X) =BP0(X)and k k
P =k kP0:
Proposition 14 ([2]). Let X be a locally convex space and P 2 C(X). Then:
1. the topology given by the norm k kP on the algebra BP(X) is …ner than the topology of uniform convergence;
2. if (Tn)n is a Cauchy sequences in(BP(X);k kP) which converges to an oper-ator T, we have T 2BP(X);
3. the algebra (BP(X);k kP) is complete if X is sequentially complete.
Proposition 15([5]). Let(X;P)be a locally convex space. An operatorT 2QP(X) is bounded in the algebra QP(X) if and only if there exists some calibration P0 2
C(X) such thatP P0 and T 2B P0(X).
De…nition 16. Let (X;P) be a locally convex space and T 2BP(X). We said that
2Cis in resolvent set (BP; T) if there exists( I T) 12BP(X). The spectral
set (BP; T) will be the complementary set of (BP; T).
Remark 17. It is obvious that we have the following inclusions
(T) (QP; T) (BP; T):
Proposition 19. Let(X;P)be a locally convex space. Then an operatorT 2QP(X)
which implies that for eachn n we have
kR( ; T)kP 0 R( ; T)
Since >0 is arbitrarily chosen, we have that
kR( ; T)kP 0<(kTkP 0) 1;(8)j j>2kTkP 0
From de…nition of normk kP it follows that
^
proposition (9) this means that
j (QP; T)j=rP(T) =1;
2
A functional calculus
In this section we assume that X will be sequentially complete locally convex space. We show that we can develop a functional for bounded elements of algebra QP(X), whereP 2C(X).
De…nition 20. Let (X;P) be a locally convex space. The Waelbroeck resolvent set of an operator T 2 QP(X), denoted by W(QP; T), is the subset of elements of
0 2C1=C[ f1g, for which there exists a neighborhood V 2V ( 0) such that:
1. the operator I T is invertible in QP(X) for all 2Vnf1g
2. the set f ( I T ) 1j 2Vnf1g g is bounded in QP(X).
The Waelbroeck spectrum of T, denoted by W(QP; T), is the complementary set of W(QP; T) in C1. It is obvious that (QP; T) W(QP; T).
De…nition 21. Let (X;P) be a locally convex space. An operator T 2 QP(X) is regular if 12= W(QP; T), i.e. there exists some t >0 such that:
1. the operator I T is invertible in QP(X), for all j j> t
2. the set f R( ; T)jj j> tg is bounded in QP(X).
Let P 2C(X) be arbitrary chosen andD C a relatively compact open set.
Lemma 22. Let p2 P then the application jf jp;D:O(D; QP(X))!R given by,
jf jp;D= sup z2D
p(f(z));(8)f 2 O(D; QP(X));
is a submultiplicative seminorm on O(D; QP(X)):
If we denote by P;D the topology de…ned by the family f jf jp;D j p 2 Pg on
O(D; QP(X)), then (O(D; QP(X)); P;D) is a l.m.c.-algebra.
Let K Cbe a compact set arbitrary chosen. We de…ne the set
O(K; QP(X)) =[fO(D; QP(X))jD C is relatively compact open g
We need the following lemma from complex analysis.
Lemma 23. For each compact set K C and each relatively compact open set
D K there exists some open set G such that:
1. K G G D;
2. Ghas a …nite number of conex components(Gi)i=1;n, the closure of which are
3. the boundary @Gi of Gi; i= 1; n, consists of a …nite positive number of closed
recti…able Jordan curves ( ij)j=1;mi, no two of which intersect;
4. K\ ij = , for each i= 1; n and every j= 1; mi.
De…nition 24. LetK andDbe like in the previous lemma. An open set Gis called Cauchy domain for pair (K; D) if it has the properties 1-4 of the previous lemma. The boundary
=[i=1;n[j=1;mi ij
of G is called Cauchy boundary for pair(K; D).
Using some results from I. Colojoara [3] we can develop a functional calculus for bounded elements of locally m-convex algebraQP(X).
Theorem 25. If P 2 C0(X) and T 2(QP(X))0, then for each relatively compact open set D W(QP; T), the application FT;D :O(D)!QP(X) de…ned by
FT;D(f) =
1 2 i
Z
f(z)R(z; T)dz;(8)f 2 O(D);
where is a Cauchy boundary for pair ( W(QP; T); D) , is a unitary continuous homomorphism. Moreover,
FT;D(z) =T;
where z is the identity function.
Like in the Banach case we make the following notation f(T) =FT;D(f).
The following theorem represents the analogous of spectral mapping theorem for Banach spaces.
Theorem 26. If P 2 C0(X), T 2 (QP(X) )0 and f is a holomorphic function on an open setD W(QP; T), then
W(QP; f(T)) =f( W(QP; T)):
Theorem 27. Assume that P 2 C0(X) and T 2 (QP(X))0. If f is holomorphic function on the open set D W(QP; T) and g 2 O(Dg), such that Dg f(D),
then(g f) (T) =g(f(T)).
Lemma 28. Assume that P 2 C0(X) and T 2 (QP(X))0. If f is holomorphic
function on the open setD W(QP; T) andf( ) =
k=1
P
k=0
k k onD, thenf(T) = k=1
P
k=0
Corollary 29. If P 2 C0(X) and T 2(QP(X))0, then expT =
k=1
P
k=0
Tk
k!.
Theorem 30. Let P 2 C0(X) and T 2 (QP(X))0. If D is an open relatively compact set which contains the set W(QP; T); f 2 O(D) and S 2(QP(X))0, such thatrP(S)<dist( W(QP; T); CnD);and T S=ST, then we have
1. W(QP; T +S) D;
2. f(T+S) = P
n 0
f(n) (T)
n! Sn.
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University of Petro¸sani,
Faculty of Sciences, Department of Mathematics, Str. Universitatii, Nr. 20, Petro¸sani,
Romania.
